- #1
Perspicacious
- 76
- 0
There is a smooth real-valued function f(x) defined on the positive reals such that, for all x>0 and for all y>0, the following identities are always true:
f(y) + 1/[(x^2)f(x)] = S/(xy)
f(x) – f(S) = y/(xS)
1/[(y^2)f(y)] - 1/[(S^2)f(S)] = x/(yS)
S is merely a function of x and y and is defined explicitly by the first equation.
It's easy to see that f(x)=1/x is one function that satisfies this system. Are there any other solutions?
f(y) + 1/[(x^2)f(x)] = S/(xy)
f(x) – f(S) = y/(xS)
1/[(y^2)f(y)] - 1/[(S^2)f(S)] = x/(yS)
S is merely a function of x and y and is defined explicitly by the first equation.
It's easy to see that f(x)=1/x is one function that satisfies this system. Are there any other solutions?